On each sheet of working write the number of the question in the top left hand corner and your name, initials and school in the top right hand corner.. • Complete the cover sheet provide[r]
(1)United Kingdom Mathematics Trust
British Mathematical Olympiad Round : Friday, 30 November 2007 Time allowed 31
2 hours
Instructions • Full written solutions - not just answers - are required, with complete proofs of any assertions you may make Marks awarded will depend on the clarity of your mathematical presentation Work in rough first, and then write up your best attempt Do not hand in rough work
• One complete solution will gain more credit than several unfinished attempts It is more important to complete a small number of questions than to try all the problems
• Each question carries 10 marks However, earlier questions tend to be easier In general you are advised to concentrate on these problems first • The use of rulers and compasses is allowed, but
calculators and protractors are forbidden
• Start each question on a fresh sheet of paper Write on one side of the paper only On each sheet of working write the number of the question in the top left hand corner and your name, initials and school in the toprighthand corner
• Complete the cover sheet provided and attach it to the front of your script, followed by your solutions in question number order
• Staple all the pages neatly together in the top left hand corner
Do not turn over untiltold to so
United Kingdom Mathematics Trust 2007/8 British Mathematical Olympiad
Round 1: Friday, 30 November 2007 Find the value of
14+ 20074+ 20084
12+ 20072+ 20082
2 Find all solutions in positive integers x, y, z to the simultaneous equations
x+y−z= 12
x2+y2−z2= 12
3 Let ABC be a triangle, with an obtuse angle atA LetQbe a point (other thanA, BorC) on the circumcircle of the triangle, on the same side of chord BC as A, and let P be the other end of the diameter throughQ.LetV andW be the feet of the perpendiculars fromQonto
CA and AB respectively Prove that the triangles P BC and AW V
are similar [Note: the circumcircle of the triangle ABC is the circle which passes through the vertices A, B andC.]
4 LetSbe a subset of the set of numbers{1,2,3, ,2008}which consists of 756 distinct numbers Show that there are two distinct elementsa, b
ofS such thata+b is divisible by
5 Let P be an internal point of triangle ABC The line through P
parallel to AB meets BC at L, the line through P parallel to BC
meetsCAat M, and the line throughP parallel to CAmeetsABat
N Prove that
BL LC ×
CM M A ×
AN N B ≤
1
and locate the position ofP in triangleABC when equality holds The function f is defined on the set of positive integers by f(1) = 1,
f(2n) = 2f(n),andnf(2n+ 1) = (2n+ 1)(f(n) +n) for alln≥1
i) Prove thatf(n) is always an integer